Exact Equation of First Order Ordinary Differential Equation

Abstract

I synthesised all the lecture/reference text notes on the exact equation of first order ordinary differential equation.

Exact Equation

Consider,

If

Thus, (1) can be rewritten,

Or, in differential form,

Equation (2) is an exact differential equation, if its left-hand side is an exact differential,


Now, from total differential,

Then, comparing (3) and (4),

Hence, (2) is exact if and only if,

Thus, equation (6) is the condition for exact equation.


Next, the integration of;

Then, differentiate ψ with respect to the opposite variable. Such that, in gradual manner,


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